Intersection Theory Class 8 Ravi
نویسنده
چکیده
1.1. Crash course in blowing up. Last time I began to talk about blowing up. Let X be a scheme, and I ⊂ OX a sheaf of ideals on X. (Technical requirement automatically satisfied in our situation: I should be a coherent sheaf, i.e. finitely generated.) Here is the “universal property” definition of blowing-up. Then the blow-up of OX along I is a morphism π : X̃ → X satisfying the following universal property. f−1IOX̃ (the “inverse ideal sheaf”) is an invertible sheaf of ideals, i.e. an effective Cartier divisor, called the exceptional divisor. (Alternatively: the scheme-theoretic pullback of the subscheme O/I is a closed subscheme of X̃ which is (effective) Cartier, and this is called the exceptional (Cartier) divisor E.) If f : Z → X is any morphism such that (fI)OZ is an invertible sheaf
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